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Supplier selection is a multi-criteria problem which includes both tangible and intangible factors. In these problems if suppliers have capacity or other different constraints two problems will exist: which suppliers are the best and how much should be purchased from each selected supplier? In this paper an integrated approach of analytic network process (ANP) and multi-objective mixed integer linear programming (MOMILP) is proposed. This integrated approach considers both tangible and intangible factors in choosing the best suppliers and defines the optimum quantities among selected suppliers to maximize the total value of purchasing (TVP), and to minimize the total cost and total defect rate and to balance the total cost among periods. The priorities are calculated for each supplier by using ANP. Four different plastic molding firms working with a refrigerator plant are evaluated according to 14 criteria that are involved in the four clusters: benefits, opportunities, costs and risks (BOCR). The priorities of suppliers will also be used as the parameters of the first objective function. This multi-objective and multi-period real-life problem is solved by using previous techniques and a reservation level driven Tchebycheff procedure (RLTP). Finally the most preferred nondominated solutions are determined by considering the decision maker's (DM's) preferences and the results obtained by these techniques are compared.

Companies need to work with different suppliers to continue their activities. In manufacturing industries the raw materials and component parts can equal up to 70% of the product cost. In such circumstances the purchasing department can play a key role in cost reduction, and supplier selection is one of the most important functions of purchasing management [9]. Several factors may affect a supplier's performance. Dickson [8] identified 23 different criteria for vendor selection including quality, delivery, performance history, warranties, price, technical capability and financial position. Hence, supplier selection is a multi-criteria problem which includes both tangible and intangible criteria, some of which may conflict. Basically there are two kinds of supplier selection problem. In the first kind of supplier selection, one supplier can satisfy all the buyer's needs (single sourcing). The management needs to make only one decision: which supplier is the best. In the second type (multiple sourcing), no supplier can satisfy all the buyer's requirements. In such circumstances management wants to split order quantities among suppliers for a variety of reasons, including creating a constant environment of competitiveness. Several methods have been proposed for single and multiple sourcing problems in the literature [9]. First publications on vendor selection can be traced back to the early 1960s. Although the problem of supplier selection is not new, quite a few researchers treat the supplier selection issue as an optimization problem, which requires the formulation of an objective function [22]. Since not every supplier selection criterion is quantitative, usually only a few quantitative criteria are included in the optimization formulation. To overcome this drawback, Ghodsypour and O’Brien [9] combined analytic hierarchy process (AHP) and linear programming to take into account tangible as well as intangible criteria and to solve order allocation problem among suppliers. Their single period model contains two different objectives: maximization of total value of purchasing (TVP) and minimization of defect rate. They used εε-constraint method to solve the problem. By following this study; Wang et al. [22] developed an integrated AHP and preemptive goal programming (PGP) approach based on multi-criteria decision-making methodology to maximize TVP and to minimize the total cost of purchasing. In Xia and Wu's paper, an integrated approach of analytical hierarchy process improved by rough sets theory and multi-objective mixed integer programming is proposed to simultaneously determine the number of suppliers to employ and the order quantity allocated to these suppliers in the case of multiple sourcing, multiple products, with multiple criteria and with supplier's capacity constraints [24]. Although timehorizon was not considered in the previous studies [7], [9], [22] and [24], time cannot be neglected in real-life problems such they are working on. To eliminate this drawback, Demirtas and Ustun [6] have also used analytic network process (ANP) and goal programming approach for multi-period lot-sizing. In this paper an integration of ANP and multi-period multi-objective mixed integer linear programming (MOMILP) is proposed to consider both tangible and intangible factors for choosing the best suppliers and define the optimum quantities among the selected suppliers. In the evaluation stage, the suppliers are evaluated according to 14 criteria that are involved in four control hierarchies: benefits, opportunities, costs and risks (BOCR). It will be useful to determine priorities by ANP, a new theory that extends the AHP. With the ANP it is recognized that there is feedback between the elements in different levels of the hierarchy and also between elements in the same level, so the decision elements are organized into networks of clusters and nodes. ANP deals systematically with all kinds of feedback and interactions (inner and outer dependence). When elements are linked only to elements in another cluster, the model shows only outer dependence. When elements are linked to the elements in their own cluster, there is inner dependence. Feedback can better capture the complex effects of interplay in human society [17]. In the shipment stage, MOMILP model is solved to obtain nondominated solutions by considering decision maker's (DM's) preferences. Over the last decade, various interactive methods and decision support systems have been developed to deal with multi-objective programming (MOP) problems [2]. In the same period, a variety of scalarization methods for finding efficient solutions of multiple objective programs MOPs have been developed. Although some of the methods work well only on problems with concave objective functions and a convex feasible region, most of the real-life problems have discrete variables, so the set of nondominated solutions for these problems is not convex. Weighted sums of the objective functions do not provide a way of reaching every nondominated solution. Besides supported nondominated solutions, there exist unsupported ones—solutions that are dominated by convex combinations of other nondominated solutions. Tchebycheff metric-based scalarizing programs have the advantage over weighted-sums programs of being able to reach, not only supported, but also unsupported nondominated solutions. A general characterization for the nondominated solution set based on the Tchebycheff metric was first proposed by Bowman [3]. A Tchebycheff scalarizing program computes the (weakly) nondominated solution closest to a reference point (e.g. the ideal criterion point) according to a (weighted) Tchebycheff (L∞)(L∞) metric. The use of reference points appeared in the early development of multiple objective programming as part of the work by Charnes and Cooper [5] on goal programming. Wierzbicki [23] produced seminal research on reference point methods, including an investigation of the characteristics of various achievement functions for allowing the search for attractive nondominated solutions to be controlled by reference points. In general, reference point approaches for multi-objective problems (considering discrete variables or not) rely on the definition of an achievement scalarizing function—as suggested by Wierzbicki [23]—by means of aspiration levels (reference point) for the objective functions. These achievement functions were designed to have a significant advantage over goal programming by producing only nondominated, or pareto-optimal, solutions. Two forms of reference points exist: aspiration points (desirable levels of achievement) and reservation points (levels of achievement that should be attained, if at all possible). In addition to their desirable structural features, reference point methods have also appeared useful from a methodological or operational perspective. They provide the “shell” for simple, decision maker controlled solution methods that do not require the decision maker to conform to any particular requirements or axioms [4]. The achievement scalarizing function projects the reference point onto the nondominated set, for instance, through the minimization of a (weighted) Tchebycheff distance to the reference point (assuming the reference point to the larger than all nondominated criterion points). General information about aspiration-based decision support systems can be found in Lewandowski and Wierzbicki [13]. Interactive methods using Tchebycheff (IWTP) or other achievement scalarizing functions have been developed for MOMILP problems by Steuer and Choo [20], Steuer [19], Karaivanova et al. [11], Karaivanova et al. [12], Vassilev and Narula [21], Narula and Vassilev [14], Reeves and MacLeod [16], Alves and Climaco [1] and [2], among others. In particular, Steuer and Choo [20] showed that by means of the lexicographic weighted Tchebycheff program every efficient solution of the nonconcave MOP is uniquely computable and all objective function values returned by this program are nondominated. An alternative approach to reducing the set of nondominated solutions within a Tchebycheff framework is a reservation level driven Tchebycheff procedure (RLTP) proposed by Reeves and MacLeod [16]. This approach uses RL based on the decision maker's responses for objective space reduction. This RLTP is shown to be more flexible than the original IWTP, while producing solutions of similar high-quality Reeves and MacLeod [16]. After a brief discussion of multi-objective programming methods, RLTP is suggested as the most suitable method to solve the multi-period MOMILP model. RLTP is more flexible than the other approaches and enables to backtrack and change previous inconsistent decisions. RLTP has a significant advantage over goal programming by producing only nondominated or pareto-optimal solutions. Finally, the most preferred nondominated solutions are obtained by RLTP regarding DM's preferences. To compare the qualities of final solutions, previous techniques are adapted to our case study. According to the results, it is shown that RLTP is better than the other methods.

In this paper, in contrast to conventional MOP techniques, a two-stage integrated approach is proposed to select suppliers and determine their shipment allocations, given a number of tangible and intangible criteria by considering time horizon. Being different from other integrated approaches, an integration of ANP and RLTP is suggested for supplier selection and multi-period shipment allocation process. In the first stage, four different suppliers that produce the plastic parts of a refrigerator plant are evaluated by using ANP. Although AHP and ANP are the multi-criteria decision-making techniques which consider both tangible and intangible factors, AHP is not sufficient to catch the effects of the same level criteria on themselves and the effects of the alternatives on the criteria. To overcome the drawbacks of AHP, although an increase in the number of pairwise comparisons and more complex calculations, priorities are calculated by using ANP. Also to improve the system consistency, real data are used for the quantitative criteria in comparisons. In the second stage, the weights computed by ANP serve as coefficients of the TVP in the MOMILP model. Because of nonconvex nature of the problem and RL, εε-constraint and RLTP methods are more suitable than goal programming techniques for the second stage. RLTP is more flexible than other Tchebycheff-based approaches and εε-constraint method. By giving DM the option of adjusting some or all of the RL, we allow them to participate more actively in the interactive solution process. In other words, DM can control objective values directly through the use of RL for objective space reduction in RLTP, rather than weight space reduction in IWTP. It is also possible to backtrack and change previous inconsistent decisions in RLTP. Although RLTP has a significant advantage over goal programming by producing only nondominated or pareto-optimal solutions, the techniques suggested for the shipment stages of integrated approaches are briefly described and applied to our MOMILP model for comparative study. The quality of final solutions obtained by εε-constraint, PGP and RLTP methods is compared by using an additive utility function. RLTP is better than the others according to the results. It is also possible to adapt this multi-period MOMILP model to multi-product case. This integrated approach can be improved to reflect DM's preferences with more accuracy.